Slice the log of goat cheese into eight even-sized pieces.
Prepare a dredging station. In one shallow bowl, whisk the egg; in a second shallow bowl, place the panko bread crumbs, flour, paprika, and 1/2 tsp salt. Dip each piece of goat cheese first in the bread crumb mixture to lightly coat, then in the egg, then in the bread crumb mixture again to coat them evenly and completely.
Place on a cookie sheet and place in the freezer to chill for at least an hour.
Drizzle at least 1/4 inch of olive oil into a frying pan and heat it to about 350 degrees F over medium-high heat. Place the chilled goat cheese rounds in the pan, and fry until golden brown underneath, 2-3 minutes. Carefully turn the rounds and cook the other side until browned. You might need to do this in batches to avoid overcrowding the pan.
Place the cooked cheese on a serving plate.
In a small saucepan over medium heat, place the grapefruit balsamic, honey, and 1/2 tsp salt. Bring to a boil and then immediately turn down the heat to low and simmer until the sauce has reduced and is slightly thickened and golden, 1-2 minutes. Immediately drizzle it over the cheese, sprinkle with paprika, and serve.
Once the fried cheese has been drizzled with the sauce it doesn't keep well, so serve immediately. However you can make it ahead of time; freeze the cheese rounds before frying. The sauce can also be made ahead and reheated immediately before serving.